Theorem looinv

Description: The Inversion Axiom of the infinite-valued sentential logic (L-infinity) of Lukasiewicz. Using dfor2 , we can see that this essentially expresses "disjunction commutes". Theorem *2.69 of WhiteheadRussell p. 108. It is a special instance of the axiom "Roll", see peirceroll . (Contributed by NM, 12-Aug-2004)

	Ref	Expression
Assertion	looinv	⊢ ( ( ( 𝜑 → 𝜓 ) → 𝜓 ) → ( ( 𝜓 → 𝜑 ) → 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		imim1	⊢ ( ( ( 𝜑 → 𝜓 ) → 𝜓 ) → ( ( 𝜓 → 𝜑 ) → ( ( 𝜑 → 𝜓 ) → 𝜑 ) ) )
2		peirce	⊢ ( ( ( 𝜑 → 𝜓 ) → 𝜑 ) → 𝜑 )
3	1 2	syl6	⊢ ( ( ( 𝜑 → 𝜓 ) → 𝜓 ) → ( ( 𝜓 → 𝜑 ) → 𝜑 ) )